FOR THE QUASI-PERIODIC SCHRÖDINGER EQUATION
by DAVIDDAMANIK and MICHAELGOLDSTEIN
ABSTRACT We study the quasi-periodic Schrödinger equation
−ψ(x)+V(x)ψ (x)=Eψ (x), x∈R
in the regime of “small” V. Let(Em,Em), m∈Zν, be the standard labeled gaps in the spectrum. Our main result says that if Em−Em≤εexp(−κ0|m|)for all m∈Zν, withεbeing small enough, depending onκ0>0 and the frequency vector involved, then the Fourier coefficients of V obey|c(m)| ≤ε1/2exp(−κ20|m|)for all m∈Zν. On the other hand we prove that if|c(m)| ≤εexp(−κ0|m|)withεbeing small enough, depending onκ0>0 and the frequency vector involved, then Em−Em≤2εexp(−κ20|m|).
CONTENTS
1. Introduction and statement of the main result . . . . 217
2. A general multi-scale analysis scheme based on the Schur complement formula . . . . 224
3. Eigenvalues and eigenvectors of matrices with inessential resonances of arbitrary order . . . . 245
4. Implicit functions defined by continued-fraction-functions . . . . 262
5. Matrices with ordered pairs of resonances . . . . 282
6. Self-adjoint matrices with a graded system of ordered pairs of resonances . . . . 300
7. Matrices with inessential resonances associated with 1-dimensional quasi-periodic Schrödinger equations . . . 324
8. Matrices with an ordered pair of resonances associated with 1-dimensional quasi-periodic Schrödinger equations . . . . 346
9. Matrices with ordered pairs of resonances associated with 1-dimensional quasi-periodic Schrödinger equations: general case . . . . 359
10. Matrices with a graded system of ordered pairs of resonances associated with 1-dimensional quasi-periodic Schrödinger equations . . . . 372
11. Proof of the main theorems . . . . 379
Acknowledgements . . . . 400
References . . . . 400
1. Introduction and statement of the main result
In the last thirty five years after the classical pioneering work [DiSi] by Dinaburg and Sinai the theory of quasi-periodic Schrödinger equations has been extensively devel- oped. Despite that there are still a number of basic problems which seem to be hard to access. Here are a few such problems:
Problem1. — Consider the Schrödinger equations (1.1) −ψ(x)+
c1cos(2πx)+c2cos(2π αx)
ψ (x)=Eψ (x), x∈R,
whereαis irrational with “nice” Diophantine properties and c1,c2are constants. Describe the eigenfunc- tions and the instability intervals of the equation.
DOI 10.1007/s10240-013-0058-x
Problem2. — Find all functions of the form
(1.2) V(x)=
n,m∈Z
c(m,n)e2πi(m+αn)x, such that the equation
(1.3) −ψ(x)+V(x)ψ (x)=Eψ (x) has the same instability intervals as Equation (1.1).
Problem3. — Give a sufficient condition for a subsetS⊂R to be the spectrum of the Equa- tion (1.3) with some function V as in (1.2).
Problem4. — Solve the KdV equation (1.4) ∂tu+∂x3u+u∂xu=0 with the initial data
(1.5) u(x,0)=c1cos(2πx)+c2cos(2π αx).
Here are two comments regarding these problems.
(1) It is known that for c1,c2 small, all generalized eigenfunctions are Floquet- like. On the other hand, for c1,c2 large, there is a collection of exponentially decaying eigenfunctions with eigenvalues which are dense in a Cantor set of positive measure. The problem is to find a method that will work for all values of c1,c2. In the discrete case, Avila has recently made significant progress in this direction in a series of papers [Av1, Av2, Av3].
(2)We state the problems for the function c1cos(2πx)+c2cos(2π αx)just for the sake of simplicity of the statement. In fact the problems are as hard for this function as for any quasi-periodic function
(1.6) V(x)=
n∈Zν
c(n)e(xnω), ω=(ω1, . . . , ων)∈Rν, nω=
njωj, e(x):=exp(2πix). In this work we study the latter case.
Let us state the main results of this work. We consider the Schrödinger equation (1.7) −ψ(x)+V(x)ψ (x)=Eψ (x), x∈R,
where V(x)is a real quasi-periodic function as in (1.6). We assume that the Fourier coef- ficients c(m)obey
(1.8) c(m)≤εexp
−κ0|m|
withεbeing small. We assume that the vectorωsatisfies the following Diophantine con- dition:
(1.9) |nω| ≥a0|n|−b0, n∈Zν \ {0} with some 0<a0<1,ν−1<b0<∞. Set (1.10)
kn= −nω/2, n∈Zν\ {0}, K(ω)= kn:n∈Zν\ {0} , Jn=
kn−δ(n),kn+δ(n)
, δ(n)=a0
1+ |n|−b0−3
, n∈Zν \ {0}, R(k)= n∈Zν\ {0} :k∈Jn
, G= k:R(k)<∞ ,
where a0,b0are as in the Diophantine condition (1.9). Let k∈Gbe such that|R(k)|>0.
Due to the Diophantine condition, one can enumerate the points of R(k) as n()(k), =0, . . . , (k), 1+(k)= |R(k)|, so that|n()(k)|<|n(+1)(k)|; see Lemma10.9in Sec- tion10. Set
(1.11)
Tm(n)=m−n, m,n∈Zν, m(0)(k)= 0,n(0)(k)
, m()(k)=m(−1)(k)∪Tn()(k)
m(−1)(k)
, =1, . . . , (k).
Theorem A. — There exists ε0 =ε0(κ0,a0,b0) >0 such that if ε≤ ε0, then for any k∈G\ ω2(Zν \ {0}), there exist E(k)∈R andϕ(k):=(ϕ(n;k))n∈Zν such that the following con- ditions hold:
(a) ϕ(0;k)=1, (1.12)
ϕ(n;k)≤ε1/2
m∈m()
exp
−7
8κ0|n−m|
, n∈/m((k))(k), ϕ(m;k)≤2, for any m∈m((k))(k).
(b) The function
ψ (k,x)=
n∈Zν
ϕ(n;k)e
x(nω+k)
is well-defined and obeys Equation (1.7) with E=E(k), that is, (1.13) Hψ (k,x)≡ −ψ(k,x)+V(x)ψ (k,x)=E(k)ψ (k,x).
(c)
(1.14)
E(k)=E(−k), ϕ(n; −k)=ϕ(−n;k), ψ (−k,x)=ψ (k,x), k02
(k−k1)2<E(k)−E(k1) <2k(k−k1)+2ε
k1<kn<k
δ(n), 0<k−k1<1/4, k1>0,
where k(0):=min(ε0,k/1024).
(d) The spectrum of H consists of the following set, S=
E(0),∞
m∈Zν\{0}:E−(km)<E+(km)
E−(km),E+(km) , where
E±(km)= lim
k→km±0,k∈G\K(ω)E(k), for km>0.
One of the central results of the current work is the following:
Theorem B. — (1)The gaps(E−(km),E+(km)) in Theorem A obey E+(km)−E−(km)≤ 2εexp(−κ20|m|).
(2)Using the notation from Theorem A, there existsε(0)such that if the gaps(E−(km),E+(km)) obey E+(km)−E−(km)≤εexp(−κ|m|)withε < ε(0),κ >4κ0, then, in fact, the Fourier coefficients c(m)obey|c(m)| ≤ε1/2exp(−κ2|m|).
Remark1.1. —(1)In a companion paper, [DG], we apply Theorem B to establish the existence of a global solution of the KdV equation
(1.15) ∂tu+∂x3u+u∂xu=0
with small quasi-periodic initial data. This application is the main objective of Theorem B. We would like to explain in this remark why the estimate in part 2 of Theorem B is crucial for the existence of a global solution of (1.15) with quasi-periodic data. Recall the following fundamental result by P. Lax [Lax]: Let u(t,x)be a function defined for 0≤t<t0, x∈R such that ∂xαu exist and are continuous and bounded in both variables for 0≤α ≤3.
Assume that u obeys Equation (1.15). Consider the Schrödinger operators (1.16) [Htψ](x)= −ψ(x)+u(t,x)ψ (x), x∈R.
Thenσ (Ht)=σ (H0)for all t. Assume that (1.17) u(t,x)=
n∈Zν
c(t,n)eixnω, with
c(t,n)≤εexp
−κ1|n|
for all 0≤t<t0,
where ε ≤ε0(a0,b0, κ1). Assume in addition that for t =0, the estimates are better:
|c(0,n)| ≤εexp(−κ0|n|),ε≤ε(0)(a0,b0, κ1). Applying Theorems A and B, one concludes that in fact|c(t,n)| ≤ε1/2exp(−κ20|n|). In other words, there is no blow up of the estimates
for the Fourier coefficients of the solution. Thus, due to Theorems A, B, to prove the ex- istence of a global solution of the KdV equation (1.15) with quasi-periodic initial data (1.18) u0(x)=
n∈Zν
c0(n)eixnω,
with |c0(n)| ≤εexp(−κ0|n|), ε≤ε(0), one only has to establish the existence of a local solution.
(2) An estimate similar in spirit to the one in the first part of Theorem B was established by Hadj Amor [HA].
(3)The problem of “keeping the exponential decay of the Fourier coefficients in check” is also well known in the KAM theory of perturbations of integrable PDE’s; see for instance the paper [EK] by Eliasson and Kuksin on periodic non-linear Schrödinger equations.
The existence of solutionsψ (k,x)as in Theorem A was discovered for a large set of k’s in the paper [DiSi] by Dinaburg and Sinai. Such solutions are called Floquet-Bloch or just Floquet solutions and the parameter k is called quasi-momentum. In [El], Eliasson proved the existence of Floquet solutions for k∈Gand also the fact that the spectrum is purely absolutely continuous.
Our approach is completely different from Eliasson’s approach. We prove exponential localization of the eigenfunctions of the dual operator. The duality underlying this approach is called Aubry duality. In [BoJi], Bourgain and Jitomirskaya used this approach to study discrete quasi-periodic Schrödinger operators for small values of the coupling constant;
see also [Bo]. Let us introduce the dual operators for (1.7). Given k∈R and a function ϕ(n), n∈Zν such that|ϕ(n)| ≤Cϕ|n|−ν−1, where Cϕ is a constant, set
(1.19) yϕ,k(x)=
n∈Zν
ϕ(n)e
(nω+k)x .
The function yϕ,k(x)satisfies Equation (1.7) if and only if (1.20) (2π )2(nω+k)2ϕ(n)+
m∈Zν
c(n−m)ϕ(m)=Eϕ(n) for any n∈Zν. Put
(1.21) h(m,n;k)=(2π )2(mω+k)2 if m=n, h(m,n;k)=c(n−m) if m=n.
Consider Hk =(h(m,n;k))m,n∈Zν.
Theorem C. — There existsε0=ε0(κ0,a0,b0) >0 such that forε≤ε0 and any k∈G\
ω
2Zν, there exists E(k)∈R andϕ(k):=(ϕ(n;k))n∈Zν such that the following conditions hold:
(1) ϕ(0;k)=1,
ϕ(n;k)≤ε1/2
m∈m()
exp
−7
8κ0|n−m|
, n∈/m((k))(k), ϕ(m;k)≤2, for any m∈m((k))(k),
(1.22)
Hkϕ(k)=E(k)ϕ(k).
(1.23) (2)
E(k)=E(−k), ϕ(n; −k)=ϕ(−n;k), (1.24)
k(0)2
(k−k1)2<E(k)−E(k1) <2k(k−k1)+2ε
k1<kn<k
δ(n)1/8
(1.25) ,
0<k−k1<1/4,k1>0, where k(0):=min(ε0,k/1024).
(3) Set E±(kn) = limk→kn±0,k∈GE(ε,k). Assume that E+(kn(0)) > E−(kn(0)). Let E−(kn(0)) <E<E+(kn(0)). Then(E−Hk)is invertible for every k.
Let us give a short description of our method and the central technical difficulty we resolve. The proof of Theorem C is built upon an abstract multi-scale analysis scheme.
We estimate the Green function (E−H)(m,n) of the matrix H:=(h(m,n;k))m,n∈, ⊂Zν moving up on the “size scale” of. This approach was introduced in the theory of Anderson localization in the fundamental work [FrSp] of Fröhlich and Spencer on ran- dom potentials and later by Fröhlich, Spencer and Wittwer in [FSW] for quasi-periodic potentials. Our multi-scale scheme is based on the Schur complement formula:
(1.26)
H1 1,2 2,1 H2
−1
=
H−11+H−111,2H˜−212,1H−11 −H1−11,2H˜−21
− ˜H−12 2,1H−11 H˜−12
, with
(1.27) H˜−21=
H2−2,1H−111,2−1
.
The main piece here isH˜−21. The iteration of (1.27) over the scales builds up a “continued-fraction- function” of the spectral parameter E and the quasi-momentum k. To estimateH˜−21on a given scale, say s, one has to study the roots of the determinant ofH2−2,1H−111,2which is the pre- vious continued-fraction-function. These roots are close to E(s−1)(k)—the eigenvalues of the matrix of the previous scale setparameterized against k. The major problem here is that E(s− 1)(k) and E(s−1)(k)can be “extremely” close for a finite (if k ∈G), but large number of times. These are the so-called essential resonances. The eigenfunctionϕ(n;k) in fact “typically” assumes values≈1 for all n∈m(0)(k); see (1.11). The sets “around”
n∈m(0)(k)produce these resonance effects. This fact gives an idea of the complexity of the central technical problem one faces in this approach. The advantage of this approach is that it eventually gives a system of equations relating the gaps in the spectrum and the Fourier coefficients. The central technical tool we develop to resolve the resonance problem consists of “continued-fraction-functions” and their roots. This is done in Sec- tion4. To give the reader an idea what this is about, consider the problem for the simplest
“continued-fraction-function”:
(1.28) E−a1(ε,k,E)− b(ε,k,E)2
u−a2(ε,k,E)=0.
The new variableεis introduced here by consideringεc(n)instead of c(n). This variable plays a crucial role since we build the solutions via analytic continuation in ε, starting at ε=0. Note that the fact that the numerator b2 here is non-negative reflects the self- adjointness of the problem, which is also crucial for the derivation. Technically, the prob- lem here is that a1and a2can be arbitrarily close due to resonances. A direct application of the implicit function theorem to
(1.29) χ (ε,k,E):=
E−a1(ε,k,E)
E−a2(ε,k,E)
−b(ε,k,E)2=0
fails (∂Eχ may have zeros). What comes to the rescue is that the symmetries in the structure of H, withbuilt appropriately, allow for the comparison
(1.30) a1(ε,k,E) >a2(ε,k,E)
for all values ofε∈(−ε0, ε0)and for k, E close to the ones in question. Due to this fact, one has two solutions E+(ε,k) >E−(ε,k). For k= −mω2 , these are very close to the two edges of the corresponding gap. One of the crucial estimates we develop says that the margins E(s2)(k)−E(s1)(k)can be estimated via the quantities|k+mω2 |. The symmetries in the structure of H withbuilt appropriately play a crucial role in this. Let us mention here that the next level “continued-fraction-functions” look as follows,
(1.31) f =f1−b21 f2
,
where f1,f2 are like in (1.28). We are interested in the solution of the equation f =0.
An important detail here is that although f1,f2 are assumed to be “small on the next scale,” their derivatives are of magnitude∼1. This accommodates the above mentioned fact that the eigenfunctionϕ(n;k)assumes values∼1 at all resonant points involved. In general the construction iterates a large number of times.
Let us say a few words about these symmetries. Given k, we define an increasing sequence(s)k of subsets ofZν,
s(s)=Zν, which allows us to analyze inductively the eigenvalues E((s)k ,k) and the eigenvectors. The construction of the sets (s)k requires involved combinatorial arguments. The set(s)k is a relatively “small” perturbation of a
union of two “large” cubes, one centered at the origin and another at n()(k); see (1.10).
The boundary of the set is of “fractal nature” built on the scale basis. The purpose of this “fractal” boundary is as follows. We need the set(s)k to be invariant under the map T(n)=n()(k)−n, and at the same time we want the boundary∂(s)k to avoid each subset m+(sk+mω) with s<s and with|E((sk+mω) ,k+mω)−E((s−1)k ,k)|being “small.”
Finally, let us say a few words about the structure of the work. First of all we split the technical difficulties into two major parts. In the first one, we develop a general the- ory of matrices which by definition have the needed structures. These matrices do not depend on the quasi-momentum k. We start with a general multi-scale analysis scheme and then inductively introduce more and more complex matrices under consideration.
This is done in Sections2–3and5–6. As already mentioned, in Section4we develop the necessary theory of “continued-fraction-functions.” In the second part, which consists of Sections7–10, we verify that the matrices dual to the quasi-periodic Schrödinger equa- tion, with appropriate(s)k , fit into the definitions from Sections2–6. Finally, in Section11 we prove the main theorems.
Remark1.2. — The fact that our presentation separates the general theory from the application to small quasi-periodic potentials with Diophantine frequency vector also has the additional benefit that in subsequent applications of the general theory, one merely needs to verify all its necessary assumptions in a given situation. We envision a number of additional applications of the general theory, such as an extension of the quasi-periodic results beyond the case of small coupling, and more generally a version of them for suit- able non-zero background potentials. We intend to address these additional applications in future works.
2. A general multi-scale analysis scheme based on the Schur complement formula
Let ⊆Zν and let H =(H(m,n))m,n∈ be an arbitrary matrix. For ⊂, denote by P the orthogonal projection onto the subspace C of all functions ψ = {ψ (n):n∈Zν} vanishing off . The restriction of H to is the operator H : C →C,
H:=PHP.
Let0⊂be an arbitrary subset and set1=\0. Then, H=P0HP0+P1HP1+P0HP1+P1HP0.
By viewingCasC1⊕C0, one has the following matrix representation of H,
(2.1) H=
H1 1,0 0,1 H0
,
where
i,j(k, )=H(k, ), k∈i, ∈j. Recall the following fact, known as the Schur complement formula.
Lemma2.1. — Let
(2.2) H=
H1 1,2 2,1 H2
,
whereHj is an (Nj ×Nj)-matrix, j =1,2, andi,j is an(Ni×Nj)-matrix. Assume that H1 is invertible. LetH˜2=H2−2,1H−111,2. Then,His invertible if and only ifH˜2is invertible; and in this case, we have
(2.3) H−1=
H−11+H−111,2H˜−212,1H−11 −H1−11,2H˜−21
− ˜H−212,1H−11 H˜−21
. Definition2.2.
(1) For each m, let γ (m) := (m) be the sequence which consists of one point m. Set (m,m;1, ):= {γ (m)},(m,n;1, ):= ∅for n=m,
(2.4)
(k, )= γ =(n1, . . . ,nk):nj ∈,nj+1=nj
, k≥2, (m,n;k, )= γ ∈(k, ),n1=m,nk=n
, m,n∈,k≥2, 1(m,n;)=
k≥1
(m,n;k, ), 1()=
m,n∈
1(m,n;).
Letγ =(n1, . . . ,nk),γ=(n1, . . . ,n), ni,nj∈Zν. Set
(2.5) γ ∪γ:=
(n1, . . . ,nk,n1, . . . ,n) if nk=n1, (n1, . . . ,nk,n2, . . . ,n) if nk=n1.
(2) Letw(m,n), D(m)be functions obeyingw(m,n)≥0, D(m)≥1, m,n∈. Forγ = (n1, . . . ,nk), set
(2.6) wD,κ0(γ ):=
1≤j≤k−1
w(nj,nj+1)
exp
1≤j≤k
D(nj)
.
Wherever we applywD,κ0(γ1∪γ2), we assume that we are in the second case in (2.5). For that matter,wD,κ0(γ1∪γ2)=wD,κ0(γ1)wD,κ0(γ2).
Let 0< κ0<1. We always assume thatw(m,m)=1 and (2.7) w(m,n)≤exp
−κ0|m−n| ,
and we set
(2.8)
WD,κ0(γ ):=exp
−κ0γ +
1≤j≤k
D(nj)
, γ :=
1≤i≤k−1
|ni−ni+1|, D(γ )¯ :=max
j D(nj).
Here,γ =0 if k=1. Obviously,wD,κ0(γ )≤WD,κ0(γ ).
(3) Let T≥8. We say thatγ =(n1, . . . ,nk), nj∈, k≥1 belongs toD,T,κ0(n1,nk;k, ) if the following condition holds:
min
D(ni),D(nj)
≤T(ni, . . . ,nj)1/5 (2.9)
for any i<j such that min
D(ni),D(nj)
≥4Tκ0−1. Note thatD,T,κ0(n1,n1;1, )= {(n1)}. Set
D,T,κ0(m,n;)=
k
D,T,κ0(m,n;k, ), D,T,κ0()=
m,n
D,T,κ0(m,n;).
(4) Set
(2.10)
sD,T,κ0;k,(m,n)=
γ∈D,T,κ0(m,n;k,)
wD,κ0(γ ), SD,T,κ0;k,(m,n)=
γ∈D,T,κ0(m,n;k,)
WD,κ0(γ ).
Note that sD,T,κ0;1,(m,m)=SD,T,κ0;1,(m,m)=exp(D(m)).
(5) Let ⊂ ¯⊂Zν. Setμ,¯(m):=dist(m,¯ \). We say that the function D(m), m∈belongs toG,,T,κ¯ 0 if the following condition holds:
(2.11) D(m)≤Tμ,¯(m)1/5 for any m such that D(m)≥4Tκ0−1.
(6) Let D ∈ G,,T,κ¯ 0. We say that γ = (n1, . . . ,nk), nj ∈ , k ≥ 1 belongs to D,T,κ0(n1,nk;k, ,R)if the following conditions hold:
(2.12) min
D(ni),D(nj)
≤T(ni, . . . ,nj)1/5
for any i<j such that min(D(ni),D(nj))≥4Tκ0−1, unless j=i+1.
Moreover, if min(D(ni),D(ni+1)) >T|(ni−ni+1)|1/5for some i, then
(2.13)
min
D(nj),D(ni)
≤T(nj, . . . ,ni)1/5, min
D(ni),D(nj)
≤T(ni, . . . ,nj)1/5, min
D(nj),D(ni+1)
≤T(nj, . . . ,ni+1)1/5, min
D(ni+1),D(nj)
≤T(ni+1, . . . ,nj)1/5, for any j<i<i+1<j.
Set D,T,κ0(m,n;,R) =
kD,T,κ0(m,n;k, ,R), D,T,κ0(,R) =
m,nD,T,κ0(m,n;,R). Given γ = (n1, . . . ,nk) ∈ D,T,κ0(n1,nk;k, ,R) \ D,T,κ0(n1,nk;k, ), we denote byP(γ )the set of all i for which min(D(ni),D(ni+1))
≥T|(ni−ni+1)|1/5. Set
(2.14)
sD,T,κ0;k,,R(m,n)=
γ∈D,T,κ0(m,n;k,,R)
wD,κ0(γ ), SD,T,κ0;k,,R(m,n)=
γ∈D,T,κ0(m,n;k,,R)
WD,κ0(γ ).
Remark2.3. —(1)Everywhere in this section the set ¯ is fixed. For this reason we suppress
¯
from the notation. We always assume in all statements that each subset⊂Zν under consideration is a subset of . The complement¯ calways means¯ \. When we apply the statements from the current section in Sections3and5, we will assume¯ =Zν. On the other hand, we will use different subsets in the role of¯ starting from Section 6. Note for that matter thatG,,T,κ¯ 0⊂G,¯1,T,κ0 if⊂ ¯1⊂ ¯.
(2)The sets of trajectoriesD,T,κ0(n1,nk;k, ),D,T,κ0(n1,nk;k, ,R)are designed to estimate the inverse for two different types of matrices we study in this work. We intro- duce these two types of matrices in Section3and Section5, respectively. We estimate the inverse via the functions
(2.15)
sD,T,κ0,ε0;(m,n):=
k≥1
εk0−1sD,T,κ0;k,(m,n), sD,T,κ0,ε0;,R(m,n):=
k≥1
εk0−1sD,T,κ0;k,,R(m,n), SD,T,κ0,ε0;(m,n):=
k≥1
εk0−1SD,T,κ0;k,(m,n), SD,T,κ0,ε0;,R(m,n):=
k≥1
εk0−1SD,T,κ0;k,,R(m,n),
respectively. One of the important properties of these functions is “functoriality” with respect to the Schur complement formula. The precise meaning of this “functoriality” is formulated in Lemma2.13. Its derivation is based on the multiplicativity property of the functionswD,κ0(γ ), WD,κ0(γ )with respect to the operationγ1∪γ2.
(3) In Sections 3–10 we will use the function WD,κ0(γ ) and the corresponding sums. We will use the functionwD,κ0(γ )and the corresponding sums in Section11.
Lemma 2.4. — Let γ =(n1, . . . ,nk)∈ D,T,κ0(n1,nk;k, ,R). Set M =4Tκ0−1, tD(γ ):= loglog MD(γ )¯ ,ϑt=
0<s≤t2−5s.
If tD(γ )≤5, then WD,κ0(γ )≤exp(−κ0γ +kM5). Otherwise, withchosen such that D(n)= ¯D(γ ), we have
WD,κ0(γ )≤
⎧⎪
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⎪⎪
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⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
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⎪⎩
e−κ0(1−ϑtD(γ ))γ+ ¯D(γ )
if, −1∈/P(γ )and maxj=D(nj) < D(n)
M2 , e−κ0(1−ϑtD(γ )+1)γ
if, −1∈/P(γ )and maxj=D(nj)≥ D(nM2), e−κ0(1−ϑtD(γ ))γ+2D(γ )¯
if∈P(γ )or−1∈P(γ )and maxj∈{/−1,}D(nj) <D(nM2), e−κ0(1−ϑtD(γ )+1)γ
if∈P(γ )or−1∈P(γ )and maxj∈{/ −1,}D(nj)≥ D(nM2). (2.16)
Here, by convention, a maximum taken over the empty set is set to be−∞.
Proof. — The verification of the estimate goes by induction in k=1,2, . . .. The estimate obviously holds for k=1. Note also that if tD(γ )≤5, the estimate holds for obvious reasons. So, we assume henceforth that tD(γ ) >5. Assume that the estimate (2.16) holds for any trajectory γ=(n1, . . . ,nt)with t ≤k−1, k≥2. Recall that is chosen such that D(n)= ¯D(γ ). There are several cases to be considered.
Case (I). Assume first that−1, /∈P(γ ). Assume also that 2≤≤k−1, so that γ1=(n1, . . . ,n−1),γ2=(n+1, . . . ,nk)are defined. Letibe such that D(ni)= ¯D(γi), i= 1,2. Note thatγ =(n1, . . . ,nk)∈D,T(n1,nk;,R)implies T(n1, . . . ,n)1/5≥D(n1), since otherwise1=−1∈P(γ ). In particular,(γ1 + |n−1−n|)1/5≥T−1D(n1)= T−1MtD(γ1)≥MtD(γ1)−1. Let us consider the following sub-cases.
(a) Assume that M2maxj<,j=1D(nj) <D(n1) <M−2D(n). This implies in par- ticular D(n1) >M2, that is, tD(γ1) >2. In particular, 4tD(γ1)−5>tD(γ1)+1. It im- plies also that tD(γ1)+2<tD(γ ). Recall that due to the inductive assumption, we have WD,κ0(γ1)≤exp(−κ0(1−ϑtD(γ1))γ1 +2D(n1)). Hence,
WD,κ0(γ1)exp
−κ0|n−1−n| (2.17)
≤exp
−κ0(1−ϑtD(γ1))
γ1 + |n−1−n|
+2D(n1)
≤exp
−κ0(1−ϑtD(γ1))
γ1 + |n−1−n| +2T
γ1 + |n−1−n|1/5
≤exp
− κ0
1−ϑtD(γ1)−2Tκ0−1
γ1 + |n−1−n|−4/5
×
γ1 + |n−1−n|
≤exp
− κ0
1−ϑtD(γ1)−2Tκ0−1M−4(tD(γ1)−1)
γ1 + |n−1−n|
≤exp
−κ0
1−ϑtD(γ1)−4−4tD(γ1)+5
γ1 + |n−1−n|
≤exp
−κ0(1−ϑtD(γ1)+1)
γ1 + |n−1−n|
≤exp
−κ0(1−ϑtD(γ ))
γ1 + |n−1−n| .
(b) Assume that D(n1)≤M−2maxj<,j=1D(nj), D(n1) <M−2D(n). Once again, tD(γ1)+2<tD(γ ). Due to the inductive assumption, this time one has WD,κ0(γ1)≤ exp(−κ0(1−ϑtD(γ1)+1)γ1). So,
WD,κ0(γ1)exp
−κ0|n−1−n| (2.18)
≤exp
−κ0(1−ϑtD(γ1)+1)
γ1 + |n−1−n|
≤exp
−κ0(1−ϑtD(γ ))
γ1 + |n−1−n| .
(c) Assume D(n1) <M−2D(n). Obviously, (a) or (b) applies. Thus, in any event, we have WD,κ0(γ1)exp(−κ0|n−1−n|)≤exp(−κ0(1−ϑtD(γ ))(γ1 + |n−1−n|)).
(d) Assume D(n1)≥ M−2D(n). Since we assumed that tD(γ ) > 5, we have D(n1) >M−2M5=M3. So, tD(γ1) >3. In particular, 4tD(γ1)−7>tD(γ1)+1. Using the inductive assumption, we obtain
WD,κ0(γ1)exp
−κ0|n−1−n| +2D(n) (2.19)
≤exp
−κ0(1−ϑtD(γ1))
γ1 + |n−1−n| +
2+2M2
D(n1)
≤exp
−κ0
1−ϑtD(γ1)−κ0−1
2+2M2
M−4(tD(γ1)−1)
×
γ1 + |n−1−n|
<exp
−κ0
1−ϑtD(γ1)−4−4tD(γ1)+7
γ1 + |n−1−n|
≤exp
−κ0(1−ϑtD(γ1)+1)
γ1 + |n−1−n|
≤exp
−κ0(1−ϑtD(γ )+1)
γ1 + |n−1−n| .
Now we prove the statement in case (I). Obviously, the cases (c) and (d) complement each other. Note also that since /∈P, one can similarly identify the cases (a)–(d) forγ2 and establish estimates similar to (2.17)–(2.19). Assume that case (c) applies for bothγ1 andγ2. Combining the estimates forγ1andγ2, we obtain the desired estimate in the first line case in (2.16). Assume now that we have case (d) forγ1and case (c) forγ2. Then,
